This is a very basic question from Rotman's Homological Algebra text. I think the issue is that I don't know how to rigorously talk about cardinality.
Problem
If $\mathcal A$ is a small category (meaning $\text{obj}(\mathcal A)$ has a cardinal number) and $F, G : \mathcal A \to \mathcal B$ are functors of the same variance, prove that $\text{Nat}(F,G)$ is a set (i.e. not a proper class).
Attempted Proof
For each $\tau \in \text{Nat}(F,G)$, there are $K := \text{card(obj}(\mathcal A))$ slots to fill for morphisms $\tau_A$, and for each such morphism there's $L := \text{card(Hom}(FA, GA))$ choices. So this set has cardinality $K * L$. $\square$