Natural Transformation of Bifunctors

Question

I had a hard time proving the statement: "a transformation between two bifunctors is natural if and only if it is a natural transformation in each of it's arguments". This is Proposition no. 2, p. 38 of Saunders Mac Lane's textbook "Categories for the Working Mathematician".

Any hints?

Thank you very much.

Answer 1

I assume by bifunctor you mean a functor $\mathcal A\times\mathcal B\to\mathcal C$.

Hint: If $f:A\to A'$ in $\mathcal A$ and $g:B\to B'$ in $\mathcal B$, then $$(f,1_{B'})\circ(1_A,g)\ =\ (f,g)\ =\ (1_{A'},g)\circ(f,1_B)\,.$$