Yoneda lemma

As a corollary of the Yoneda lemma, given a pair of contravariant hom functors #92;mbox#123;Hom#125;(-,A) and #92;mbox#123;Hom#125;(-,B), then any natural transformation #92;alpha from #92;mbox#123;Hom#125;(-,A) to #92;mbox#123;Hom#125;(-,B) is determined by the choice of some function f#58;A#92;rightarrowB to map the identity #92;mbox#123;id#125;#95;A#58;A#92;rightarrowA to, by the component #92;alpha#95;A#58;#92;mbox#123;Hom#125;(A,A)#92;rightarrow#92;mbox#123;Hom#125;(A,B) of #92;alpha. This implies that the Yoneda functor is fully faithful, which in turn implies that Yoneda embeddings are possible.

And then there's the Yoneda Lemma embodied in the classical Gaussian row reduction operation, that a given row reduction operation (on matrices with say k rows) being a "natural" operation (in the sense of natural transformations) is just multiplication (on the appropriate side) by the effect of that operation on the k-by-k identity matrix. And dually for column-reduction operations :-)

2020, Emily Riehl, quoting Fred E. J. Linton, The Yoneda lemma in the category of Matrices