Charles Ehresmann had a natty way of developing natural transformations. For a category $C$ let $\square C$ be the double category of commuting squares in $C$. Then for a small category $B$ we can form Cat($B,\square_1 C$), the functors from $B$ to the direction 1 part of $\square C$. This gets a category structure from the category structure in direction 2 of $\square C$. So we get a category CAT($B,C$) of functors and natural transformations. This view makes it easier to verify the law

Cat($ A \times B,C) \cong $Cat($ A, $CAT($B,C$)).

And this method goes over to topological categories as well:

R. Brown and P. Nickolas, ``Exponential laws for topological categories, groupoids and groups and mapping spaces of colimits'', {\em Cah. Top. G'eom. Diff.} 20 (1979) 179-198.

See also Section 6.5 of my book `Topology and groupoids' for using the homotopy terminoloby for natural equivalences.

Charles Ehresmann had a natty way of developing natural transformations. For a category $C$ let $\square C$ be the double category of commuting squares in $C$. Then for a small category $B$ we can form Cat($B,\square_1 C$), the functors from $B$ to the direction 1 part of $\square C$. This gets a category structure from the category structure in direction 2 of $\square C$. So we get a category CAT($B,C$) of functors and natural transformations. This view makes it easier to verify the law

Cat($ A \times B,C) \cong $Cat($ A, $CAT($B,C$)).

And this method goes over to topological categories as well:

R. Brown and P. Nickolas, ``Exponential laws for topological categories, groupoids and groups and mapping spaces of colimits'', Cah. Top. G'eom. Diff. 20 (1979) 179-198.

See also Section 6.5 of my book Topology and Groupoids for using the homotopy terminology for natural equivalences, as it was in the first 1968 edition entitled "Elements of Modern Topology" (McGraw Hill).