Exact sequence via $[-,-]_*$ with HEP

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I was wondering if it is possible to identify the sequence

$$[C(f),Z]_* \rightarrow[Y,Z]_* \rightarrow [X,Z]_* $$

induced by the sequence of based maps between pointed spaces $$ X \rightarrow Y \hookrightarrow C(f)$$ if the pair (Y,X) has HEP.