Suppose for the moment that $C$ is small, so that $[C,Set]$ is locally small. Given two functors $C\times [C,Set]\to Set$: the first sends an object $(A,F)$ to $FA$, and a morphism $(A\stackrel{f}{\to }{A}^{\prime },F\stackrel{\alpha }{\to }{F}^{\prime })$ to the diagonal of The second is the composite

$$C\times [C,Set]\stackrel{Y\times 1}{\to }{[C,Set]}^{op}\times [C,Set]\stackrel{[C,Set](\bullet ,\bullet )}{\to }Set$$

where $Y$ is a Yoneda embedding. Then $\mathrm{\Phi }$ and $\mathrm{\Psi }$ define a natural isomorphism between these two.

In elementary terms, this says that if $x\in FA$, and $x^{\prime }\in F^{\prime }{A}^{\prime }$ is its image under the diagonal, then $\mathrm{\Psi }(x^{\prime })$ is the composite

$$C(A^{\prime },\bullet )\stackrel{C(f,\bullet )}{\to }C(A,\bullet )\stackrel{\mathrm{\Psi }(x)}{\to }F\stackrel{\alpha }{\to }{F}^{\prime }.$$

This makes sense without the assumption that $C$ is small, and it’s true since the composite maps

$$1_{A^{\prime }}\mapsto f\mapsto (Ff)(x)\mapsto {\alpha }_{A^{\prime }}(Ff)(x).\square$$