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Let $X$ and $Y$ and $Z$ be compactly generated Hausdorff spaces. Is it true that the composition map $Z^Y \times Y^X \rightarrow Z^X$ is continuous? Here the function spaces are given the compact-open topology. If it's necessary, I'd be happy to $k$-ify everything in sight to ensure that all spaces I have written down are compactly generated.

I know this works if $Y$ is locally compact (with no further assumptions on $X$ and $Z$, at least if you define "local compactness" correctly), but I can't figure out if you can generalize this to the compactly generated setting.

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Yes, this is true.

This is precisely Theorem 8.18 in Gray's Homotopy Theory, An Introduction to Algebraic Topology (1975).

Note that this is in terms of "$k$-fications" as he uses the convention $Y^X := kC(X,Y)$ (Definition 8.14).

The whole Chapter 8 is a very convenient exposition of CGH spaces :-)

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