If F and G are ( covariant ) functors between the categories C and D, then a natural transformation η from F to G associates to every object X in C a morphism in D such that for every morphism in C, we have ; this means that the following diagram is commutative:

If we set, then η is continuous at 0.

If F and G are functors between the categories C and D, then a natural transformation η from F to G associates to every object X in C a morphism between objects of D, called the component of η at X, such that for every morphism we have:

If, for every object X in C, the morphism η < sub > X </ sub > is an isomorphism in D, then η is said to be a ( or sometimes natural equivalence or isomorphism of functors ).

If η ( θ ) = θ, then the exponential family is said to be in canonical form.

Let P be a principal H-bundle on M, equipped with a Cartan connection η: TP → g. If g is a reductive module for H, meaning that g admits an Ad ( H )- invariant splitting of vector spaces g = h ⊕ m, then the m-component of η generalizes the solder form for an affine connection.

If the continuum hypothesis holds, all real closed fields with cardinality the continuum and having the η < sub > 1 </ sub > property are order isomorphic.

If the circumstances are faced frankly it is not reasonable to expect this to be true.

If his dancers are sometimes made to look as if they might be creatures from Mars, this is consistent with his intention of placing them in the orbit of another world, a world in which they are freed of their pedestrian identities.

If a work is divided into several large segments, a last-minute drawing of random numbers may determine the order of the segments for any particular performance.

If they avoid the use of the pungent, outlawed four-letter word it is because it is taboo ; ;

If Wilhelm Reich is the Moses who has led them out of the Egypt of sexual slavery, Dylan Thomas is the poet who offers them the Dionysian dialectic of justification for their indulgence in liquor, marijuana, sex, and jazz.

If he is the child of nothingness, if he is the predestined victim of an age of atomic wars, then he will consult only his own organic needs and go beyond good and evil.

If it is an honest feeling, then why should she not yield to it??

If he thus achieves a lyrical, dreamlike, drugged intensity, he pays the price for his indulgence by producing work -- Allen Ginsberg's `` Howl '' is a striking example of this tendency -- that is disoriented, Dionysian but without depth and without Apollonian control.

If love reflects the nature of man, as Ortega Y Gasset believes, if the person in love betrays decisively what he is by his behavior in love, then the writers of the beat generation are creating a new literary genre.

If he is good, he may not be legal ; ;

If the man on the sidewalk is surprised at this question, it has served as an exclamation.

If the existent form is to be retained new factors that reinforce it must be introduced into the situation.

If we remove ourselves for a moment from our time and our infatuation with mental disease, isn't there something absurd about a hero in a novel who is defeated by his infantile neurosis??

If many of the characters in contemporary novels appear to be the bloodless relations of characters in a case history it is because the novelist is often forgetful today that those things that we call character manifest themselves in surface behavior, that the ego is still the executive agency of personality, and that all we know of personality must be discerned through the ego.

If he is a traditionalist, he is an eclectic traditionalist.

If our sincerity is granted, and it is granted, the discrepancy can only be explained by the fact that we have come to believe hearsay and legend about ourselves in preference to an understanding gained by earnest self-examination.

If to be innocent is to be helpless, then I had been -- as are we all -- helpless at the start.

If your pool is located on or near sloping ground, it may have natural drainage which is certainly more desirable than to be faced with the annual expense and labor of first pumping out the water and then scooping out all the debris.

If this aspect of death as punishment is not distinguished from the idea of death as natural termination, the conclusion seems inevitable that temporal existence itself is a form of punishment rather than the state into which man is put by the will of the Creator.

** If the set A is infinite, then there exists an injection from the natural numbers N to A ( see Dedekind infinite ).

If the partnership's preceding call is a natural bid in notrump, then 4NT is also natural.

* Indicative conditional, a conditional in the form of " If A then B " in natural languages

" 9: 2 If the passage in Isaiah is interpreted literally, a return to the vegetarian diet of Eden seems to be a natural conclusion. Gen 1: 29-30

If a 12 is rolled, it is a push ( this is to reverse the natural house advantage: because the don't come bet puts the player in the position of the house, it would otherwise favour the player ).

If not " true " crannogs, small occupied islets ( often at least partially artificial in nature ) may be referred to as island duns, although rather confusingly, 22 islet-based sites are classified as ' proper ' crannogs due to the different interpretations of the inspectors or excavators who drew up field reports Canmore search for crannog in the Western Isles Hebridean island dwellings or crannogs were commonly built on both natural and artificial islets, usually reached by means of a stone causeway.

:" If I had stated ... the possibility of the introduction or origination of fresh species being a natural, in contradistinction to a miraculous process, I should have raised a host of prejudices against me, which are unfortunately opposed at every step to any philosopher who attempts to address the public on these mysterious subjects ".

If X is a topological space, there is a natural way of transforming X /~ into a topological space ; see quotient space for the details.

If m and n are natural numbers and f ( x ) is a smooth ( meaning: sufficiently often differentiable ) function defined for all real numbers x in the interval, then the integral

If G = GL < sub >*</ sub >( K ), then the set of natural numbers is a proper subset of G < sub > 0 </ sub >, since for each natural number n, there is a corresponding identity matrix of dimension n. G ( m, n ) is empty unless m = n, in which case it is the set of all nxn invertible matrices.

# If the noun is animate, natural gender tends to dictate grammatical gender.

If the player ascends the first eight strings of the bass bridge, they will encounter a flatted seventh ( C natural in this case ), because this note is drawn from the G tetrachord.

If the ham has less than 20. 5 % but is at least 18. 5 % protein, it can be called " ham with natural juices ".

Since counter-clockwise may be defined in terms of up, forward, and right, this experiment unambiguously differentiates left from right using only natural elements: If they were reversed, or the atoms spun clockwise, the radiation would follow the spin axis instead of being opposite to it.

# If a and p are natural numbers such that a < sup > p </ sup > − 1 is prime, then a

If that ideal size is large enough to supply the whole market, then that market is a natural monopoly.

If ( V, ‖·‖) is a normed vector space, the norm ‖·‖ induces a metric ( a notion of distance ) and therefore a topology on V. This metric is defined in the natural way: the distance between two vectors u and v is given by ‖ u − v ‖.

Today, biologists denounce the Naturalistic Fallacy because they want to describe the natural world honestly, without people deriving morals about how we ought to behave — as in: If birds and beasts engage in adultery, infanticide, cannibalism, it must be OK ).

If the key signature indicates G-sharp, a local flat before a G makes it G-flat ( not G natural ), though often this type of rare accidental is expressed as a natural, followed by a flat () to make this clear.