Are PL maps determined by PL-paths?

Let $X,Y$ be polyhedra. If $f\colon X\rightarrow Y$ is a PL map and $g\colon I\rightarrow X$ is a PL map (where $I$ denotes the interval), then $f\circ g$ is a PL map. Is the converse true? i.e.

If $X,Y$ are polyhedra and $f\colon X\rightarrow Y$ is a function of the underlying sets such that for all PL map $g\colon I\rightarrow X$, $f\circ g$ is PL, is it true that $f$ is PL?

Also: Which limits and colimits does the PL category have?

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