Arrows, furnished by Yoneda

Question

What are some examples of 'important arrows' in a category that are significantly easier to define via fullness of the Yoneda embedding than in the base category?

The example that brought this to mind is the internal matrix multiplication arrow$$R^{n\times k}\times R^{k\times m}\to R^{n\times m}$$ for a ring object $R$ in a category $\mathcal{C}$ with products and a terminal object ${\bf 1}$. By Yoneda we can just define functions $$f_U:{\bf Hom}_\mathcal{C}(U,R^{n\times k}\times R^{k\times m})\to{\bf Hom}_\mathcal{C}(U,R^{n\times m})$$ $$\big(\ [x_{ab}]_{\substack{a<n \\ b<k}},[y_{cd}]_{\substack{c<k \\ d<m}}\big)\mapsto[\Sigma_{a<k}x_{ia}y_{aj}]_{\substack{i<n \\ j<m}}$$ as usual and check that they're natural with respect to precomposition in $\mathcal{C}$ to obtain a natural transformation $$f:{\bf Hom}_\mathcal{C}(-,R^{n\times k}\times R^{k\times m})\Rightarrow{\bf Hom}_\mathcal{C}(-,R^{n\times m})$$ and take $$f_{R^{n\times k}\times R^{k\times m}}(1_{R^{n\times k}\times R^{k\times m}}):R^{n\times k}\times R^{k\times m}\to R^{n\times m},$$ as the desired arrow, whereas without Yoneda the definition is impractically long (as far as I can tell).

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EDIT: Below is an expanded explanation of what is gained above, and a suggestion for where other examples might exist.

For any category $\mathcal{C}$ and object $X\in{\bf Ob}_\mathcal{C}$, define the class of generalized elements of $X$ $${\bf Hom}_\mathcal{C}(\ \ ,X):=cod^{-1}(X)\subseteq{\bf Hom}_\mathcal{C} \ \ \Big(=\bigcup_{W\in{\bf Ob}_\mathcal{C}}{\bf Hom}_\mathcal{C}(W,X)\Big).$$ For objects $X,Y\in{\bf Ob}_\mathcal{C}$, a function $$f:{\bf Hom}_\mathcal{C}(\ \ ,X)\to{\bf Hom}_\mathcal{C}(\ \ ,Y)$$ between the classes of generalized elements for two objects corresponds to an arrow $$f':X\to Y$$ between the underlying objects iff $f$ respects domains and is linear with respect to precomposition; that is, iff for all $h\in{\bf Hom}_\mathcal{C}(\ \ ,X)$ and $g\in{\bf Hom}_\mathcal{C}(\ \ ,dom(h))$ we have $$dom(f(h))=dom(h),$$ $$f(h\circ g)=f(h)\circ g.$$ (This is just a repackaging of disjoint action on hom-classes and naturality.) These conditions are trivially satisfied by the postcomposition functions induced by arrows in $\mathcal{C}$, but they are also pretty trivially satisfied by many other definitions for functions between classes of generalized elements. In particular the definition of $f$ above the line trivially respects domains, and linearity with respect to precomposition amounts to showing that for any arrow $g:A\to B$ and any arrow $([h_{ij}]_{\substack{i< n \\ j<k}},[h'_{ij}]_{\substack{i<k \\ j<m}}):B\to R^{m\times k}\times R^{k\times n}$ we have $$(\sum_{\ell<k}h_{i\ell}h'_{\ell j})\circ g=\sum_{\ell<k}(h_{i\ell}\circ g)(h'_{\ell j}\circ g)$$ for all $i<n$ and $j<m$, which in turn basically amounts to showing that $$(hh')\circ g=(h\circ g)(h'\circ g)$$ and $$(h+h')\circ g=(h\circ g)+(h'\circ g)$$ for any arrow $g:A\to B$ and any generalized elements $h,h':B\to R$, which are both completely trivial to see.

Note further that by co-Yoneda, functions between co-generalized elements which respect codomains and are linear with respect to postcomposition correspond to arrows in the opposite direction between the underlying objects.

I'm pretty sure this is just the dual side of the 'internal language of a category' approach to proving commutativity for diagrams, where the diagram chase would otherwise be giant and ugly. That uses faithfulness of Yoneda, whereas this uses fullness.

For matrix multiplication without the Yoneda route, we need to start with $R^{n\times k}\times R^{k\times m}$, copy the left factor $m$ times and the right factor $n$ times, shuffle the coordinates around appropriately, collapse back down with pointwise multiplication, then sum along rows (or columns, depending on how you do the coordinate shuffle). We would then still have to prove unitarity and associativity for this version of the definition. (yikes!)

Other settings where the internal language of a category is heavily used probably admit further examples, so topos theory or SDG seem like good candidates.

Answer 1

An example similar in spirit to yours is giving explicit examples of affine group schemes. Take, for example, $GL_n$: if we wanted to work solely in $\text{Aff} = \text{CRing}^{op}$ we'd have to write down a comultiplication and antipode on $\mathbb{Z}[x_{ij}][\det^{-1}]$, check the Hopf algebra axioms, etc. Instead we can argue using the Yoneda embedding, which is to say we can work with the functor of points $R \mapsto GL_n(R)$. Then we just have to check that $GL_n(R)$ is always a group and this group structure is natural in $R$. This boils down to the naturality of matrix multiplication as in your example.

A related funny example is the "affine ring scheme" $R \mapsto M_n(R)$, from which we can obtain $GL_n(R)$ as the group of units. Working solely in $\text{Aff}$ corresponds to writing down a commutative ring $\mathbb{Z}[x_{ij}]$ equipped with two comultiplications, one for the addition and one for the multiplication in $M_n(R)$, and checking the Hopf algebra axioms for one, the bialgebra axioms for the other, then some awful distributivity axiom between them. Meanwhile checking that $M_n(R)$ is a ring and that this ring structure is natural in $R$ is very straightforward.