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An explicit proof can be found in Cole's Many Homotopy Categories Are Homotopy Categories (Lemma 3.4). Some topological properties of the interval are used and it seems crucial to the argument. These properties are essentially that $I \times I$ retracts onto both $I \times \{ 0 \}$ and $I \times \{ 0 \} \cup \{ 0, 1 \} \times I$ and that $(I \times I, I \times \{ 0 \})$ and $(I \times I, I \times \{ 0 \} \cup \{ 0, 1 \} \times I)$ are homeomorphic as pairs of spaces.

This doesn't hold e.g. for the simplicial interval and it seems that constructing Hurewicz type model structures on simplicial categories is much more subtle. Some theory for pretty general enrichments is developed in Strong Cofibrations and Fibrations in Enriched Categories by Schwänzl and Vogt.

EDIT: I have realized that I misread your question. You were asking about the projection $X^I \to X$ while the above applies to $X^I \to X \times X$ which is more subtle. With Cole's definition of strong fibrations it is obvious that $X^I \to X$ is one. May and Sigurdsson use an a priori stronger definition and I suspect that proving that theses two definitions agree may involve arguments like Cole's Lemma 3.4 mentioned above.

An explicit proof can be found in Cole's Many Homotopy Categories Are Homotopy Categories (Lemma 3.4). Some topological properties of the interval are used and it seems crucial to the argument. These properties are essentially that $I \times I$ retracts onto both $I \times \{ 0 \}$ and $I \times \{ 0 \} \cup \{ 0, 1 \} \times I$ and that $(I \times I, I \times \{ 0 \})$ and $(I \times I, I \times \{ 0 \} \cup \{ 0, 1 \} \times I)$ are homeomorphic as pairs of spaces.

This doesn't hold e.g. for the simplicial interval and it seems that constructing Hurewicz type model structures on simplicial categories is much more subtle. Some theory for pretty general enrichments is developed in Strong Cofibrations and Fibrations in Enriched Categories by Schwänzl and Vogt.

EDIT: I have realized that I misread your question. You were asking about the projection $X^I \to X$ while the above applies to $X^I \to X \times X$ which is more subtle. With Cole's definition of strong fibrations it is obvious that $X^I \to X$ is one. May and Sigurdsson use a slightly different definition but Cole proves that they are actually equivalent (see my comments below).

An explicit proof can be found in Cole's Many Homotopy Categories Are Homotopy Categories (Lemma 3.4). Some topological properties of the interval are used and it seems crucial to the argument. These properties are essentially that $I \times I$ retracts onto both $I \times \{ 0 \}$ and $I \times \{ 0 \} \cup \{ 0, 1 \} \times I$ and that $(I \times I, I \times \{ 0 \})$ and $(I \times I, I \times \{ 0 \} \cup \{ 0, 1 \} \times I)$ are homeomorphic as pairs of spaces.

This doesn't hold e.g. for the simplicial interval and it seems that constructing Hurewicz type model structures on simplicial categories is much more subtle. Some theory for pretty general enrichments is developed in Strong Cofibrations and Fibrations in Enriched Categories by Schwänzl and Vogt.

An explicit proof can be found in Cole's Many Homotopy Categories Are Homotopy Categories (Lemma 3.4). Some topological properties of the interval are used and it seems crucial to the argument. These properties are essentially that $I \times I$ retracts onto both $I \times \{ 0 \}$ and $I \times \{ 0 \} \cup \{ 0, 1 \} \times I$ and that $(I \times I, I \times \{ 0 \})$ and $(I \times I, I \times \{ 0 \} \cup \{ 0, 1 \} \times I)$ are homeomorphic as pairs of spaces.

This doesn't hold e.g. for the simplicial interval and it seems that constructing Hurewicz type model structures on simplicial categories is much more subtle. Some theory for pretty general enrichments is developed in Strong Cofibrations and Fibrations in Enriched Categories by Schwänzl and Vogt.

EDIT: I have realized that I misread your question. You were asking about the projection $X^I \to X$ while the above applies to $X^I \to X \times X$ which is more subtle. With Cole's definition of strong fibrations it is obvious that $X^I \to X$ is one. May and Sigurdsson use an a priori stronger definition and I suspect that proving that theses two definitions agree may involve arguments like Cole's Lemma 3.4 mentioned above.

An explicit proof can be found in Cole's Many Homotopy Categories Are Homotopy Categories (Lemma 3.4). Some topological properties of the interval are used and it seems essential to the argument. These properties are essentially that $I \times I$ retracts onto both $I \times \{ 0 \}$ and $I \times \{ 0 \} \cup \{ 0, 1 \} \times I$ and that $(I \times I, I \times \{ 0 \})$ and $(I \times I, I \times \{ 0 \} \cup \{ 0, 1 \} \times I)$ are homeomorphic as pairs of spaces.

This doesn't hold e.g. for a simplicial interval and it seems that constructing similar model structures on simplicial categories is much more subtle. Some theory for pretty general enrichments is developed in Strong Cofibrations and Fibrations in Enriched Categories by Schwänzl and Vogt.

An explicit proof can be found in Cole's Many Homotopy Categories Are Homotopy Categories (Lemma 3.4). Some topological properties of the interval are used and it seems crucial to the argument. These properties are essentially that $I \times I$ retracts onto both $I \times \{ 0 \}$ and $I \times \{ 0 \} \cup \{ 0, 1 \} \times I$ and that $(I \times I, I \times \{ 0 \})$ and $(I \times I, I \times \{ 0 \} \cup \{ 0, 1 \} \times I)$ are homeomorphic as pairs of spaces.

This doesn't hold e.g. for the simplicial interval and it seems that constructing Hurewicz type model structures on simplicial categories is much more subtle. Some theory for pretty general enrichments is developed in Strong Cofibrations and Fibrations in Enriched Categories by Schwänzl and Vogt.