An exercise about bifunctors from Riehl's "Category Theory in Context" This is an exercise from E.Riehl's book "Category Theory in Context" (p.48, ex.1.7.vii)

Prove that a bifunctors $F\colon\mathsf{C}\times\mathsf{D}\to\mathsf{E}$ determines and is uniquely determined by:

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*A functor $F(c,-)\colon\mathsf{D}\to\mathsf{E}$ for each $c \in \mathsf{C}$,


*A natural transformation $F(f,-)\colon F(c,-) \Rightarrow F(c',-)$ for each $f\colon c\to c'$, defined functorially in $\mathsf{C}$.

I'm having trouble figuring out two things regarding this exercise:

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*What is the functor $F(c,-)$ for a fixed $c \in \mathsf{C}$? Clearly, it maps each $d \in \mathsf{D}$ to $F(c,d)$, but to what it maps a morphism $g\colon d\to d'$ in $\mathsf{D}$? To $F(1_{X},g)?$


*Given an aformentioned natural transformation $F(f,-)\colon F(c,-)\Rightarrow F(c',-)$, what are it's components? That is, given $d \in \mathsf{D}$, what is a morphism $F(f,-)_d\colon F(c,d)\to F(c',d)$?

Yes, to your first question, or more precisely to $F(1_c,g)$
where $1_c$ is the identity morphism on $c$.
The morphisms you seek in your second are the $F(f,1_d)$.
