This question arose from the first pages of Delignes Equations Differentielles.
There he defines a connection on a vector bundle $V$ on an complex analytic space $X$ as the following Data:
For all analytic spaces $S$ and pairs $(x,y)$ of $S$-points of $X$ (i.e. Morphisms from $S$ to $X$), such that the morphism $(x,y):S\longrightarrow X\times X$ factors through the first infinitesimal neighbourhood $X_1$ of the diagonal $X_0$: a morphism $\gamma_{x,y}:x^*V\longrightarrow y^*V$ s.t.:
- For all morphisms $f:T\longrightarrow S$ we have $f^*\gamma_{x,y}=\gamma_{xf,yf}$
- $\gamma_{x,x}=\operatorname{id}$
He then claims a little bit further that this is the same as a homomorphism $\gamma:p_1^*V\longrightarrow p_2^*V$ which induces the identity on $X_0$ and that this is automatically an isomorphism. (Here the $p_i$ denote the projections of $X_1$ to $X$).
For me this sounds like he uses the Yoneda Lemma here to derive this claim, somehow interpreting the definition of a connection as a natural transformation between a representable functor (probably represented by $X_1$?) and another one. I played around with this idea but wasn't able to make it precise, so my question is: Can we, and if yes how, interpret the claim he makes as an application of the Yoneda Lemma?