I'm having a hard time understanding the following excerpt which appears on page 38 of Mac Lane's Categories for the Working Mathematician:
Next consider natural transformations between bifunctors $S, S': B \times C \to D$. Let $\alpha$ be a function which assigns to each pair of objects $b \in B, c \in C$ an arrow $$ \alpha(b,c): S(b, c) \to S'(b, c) $$ in $D$. Call $\alpha$ natural in $b$ if for each $c \in C$ the components $\alpha(b, c)$ for all $b$ define $$ \alpha(-,c): S(-,c) \to S'(-,c) $$ a natural transformation of functors $B \to D$.
I understand the concept of a natural transformation from one functor to another. But the definition of "natural" is confusing me. My interpretation of the above definition so far is the following: $\alpha$ is "natural in $C$" if, for all $b \in B$, $$ \alpha(b, -) : S(b, -) \to S'(b, -) $$ is a natural transformation of functors which are going from $C \to D$.
However, I feel like that might not be what Mac Lane is trying to say here. In addition, I am confused by the use of the word "components" here. I understand what a component is, as on page 16, Mac Lane defines "components" within the definition of a natural transformation; but here, he uses the word "components" before introducing the context of a natural transformation.
Perhaps someone can explain the definition of $\alpha$ better than Mac Lane and clarify on what he means by components here?
Apologies in advance if this is a duplicate question, as I made sure to search thoroughly before asking this.
It seems that you do more or less understand what's going on. A natural transformation $\alpha: F\to G$ where $F,G:\mathcal C\to\mathcal D$ is a family of arrows of $\mathcal D$ indexed by objects of $\mathcal C$ which satisfies some laws (i.e. the naturality laws). It's perfectly reasonable to refer to the elements of an arbitrary family of arrows as components. Indeed, when $\mathcal C$ is small (which, for the set theoretic foundations Mac Lane is using, is always the case with respect to some universe), then $\mathsf{Nat}(F,G)$, the set of natural transformations from $F$ to $G$, is a subset of the product $\prod_{C\in\mathsf{Ob}(\mathcal C)}\mathcal D(FC,GC)$. The components of a natural transformation are exactly the components of this product. This is just the common use of the term "component" in mathematics in general.
Other than the quote talking about the $\mathcal B$ component (of the product of categories $\mathcal B\times \mathcal C$), that is what Mac Lane's saying. For the $\mathcal C$ component, that's exactly what Mac Lane intends.