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For questions about or involving cofibrations which are maps which satisfy the homotopy extension property for all spaces.

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When is the inclusion of a relative mapping space into a mapping space a cofibration?

Suppose the inclusions $A\hookrightarrow X$ and $B\hookrightarrow Y$ are cofibrations, would that be enough to ensure the inclusion $\operatorname{Map}(X,A;Y,B)\hookrightarrow\operatorname{Map}(X,Y)$ is …