product functor of topological spaces for a locally compact space as left adjoint

Let \((X, \mathcal{T})\) be a locally compact topological space. Then the endofunctor

\begin{align*} X \times - : \mathrm{Top} \rightarrow& \mathrm{Top} \\ Y \mapsto& (X \times Y) \\ (f: Z \rightarrow Y) \mapsto& ((\mathrm{id}_{X}, f): X \times Z \rightarrow X \times Y) \end{align*}

is a left adjoint to the covariant compact open functor \(\mathrm{Map}(X,-)\)

Consider

\begin{align*} \mathrm{comp}: \mathrm{Hom}_{\mathrm{Top}}((X \times - ),-) \rightarrow& \mathrm{Hom}_{\mathrm{Top}}(-, \mathrm{Map}(X, -)) \\ \end{align*}

where for \(Y,Z\)

\begin{align*} \mathrm{comp}: \mathrm{Map}_{\mathrm{Top}}((X \times Y),Z) \cong \mathrm{Map}_{\mathrm{Top}}(Y, \mathrm{Map}(X,Z)) \end{align*}

is a Bijection (cf. currying of mapping spaces bijective for locally compact spaces)

Thus by natural isomorphism and isomorphism on the objects (since hom-sets are in category set) we conclude, that \(\mathrm{comp}\) is a natural isomorphism, hence \(X \times -\) a left adjoint