compact open bifunctor

Given category Top and the product category with the opposite category, the compact open bifunctor is defined as

\begin{align*} \mathrm{Map}: \mathrm{Top}^{\mathrm{op}} \times \mathrm{Top} \rightarrow& \mathrm{Top} \\ (X,Y) \mapsto& \mathcal{C}(X,Y) \\ (f: X' \rightarrow X, g: Y \rightarrow Y') \mapsto& (g^{ \leftarrow} \circ f^{\rightarrow}: \mathcal{C}(X,Y) \rightarrow \mathcal{C}(X',Y')) % g_* \circ - \circ f^* \end{align*}

where \(g^{ \leftarrow}\) is the postcomposition, \(f^{ \rightarrow}\) is the precomposition and \(\mathcal{C}(X,Y)\) is equipped with the compact open topology