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Actually, you can, and you don't need accessibility (local presentability of the underlying category is enough).

Under your assumption, for each $i:A \to B$ a generating cofibration, take $B \coprod_A B \hookrightarrow I_A B \to B$ a cylinder object, and let $j_i : B \hookrightarrow I_A B$ be the first leg inclusion. Then the $j_i$ form a generating set of trivial cofibrations.

The proof essentially use that all objects are fibrant, and that you can consider the weak factorization system generated by the $j_i$ (so some smallness or local presentability assumption).

The key step in the proof is to observe that a map that has the RLP against the $j_i$ and is a weak equivalence is a trivial fibration. This easily follows from the fact that weak equivalences between fibrant objects have the "up to homotopy" lifting property against all cofibration and that lifting property against the $j_i$ is enough to rectify this in an actual lifting property. One then check by hand the cofibration, weak equivalence and the map with the lifting property against the $j_i$ forms a model structure, with the same cofirbation and weak equivalences as the one you started from, hence is the the same model structure.

I found it explicitly written out as corollary 3.2 of this paper of Valery Isaev.

Actually, you can, and you don't need accessibility (local presentability of the underlying category is enough).

Under your assumption, for each $i:A \to B$ a generating cofibration, take $B \coprod_A B \hookrightarrow I_A B \to B$ a cylinder object, and let $j_i : B \hookrightarrow I_A B$ be the first leg inclusion. Then the $j_i$ form a generating set of trivial cofibrations.

The proof essentially use that all objects are fibrant, and that you can consider the weak factorization system generated by the $j_i$ (so some smallness or local presentability assumption).

The key step in the proof is to observe that a map that has the RLP against the $j_i$ and is a weak equivalence is a trivial fibration. This easily follows from the fact that weak equivalences between fibrant objects have the "up to homotopy" lifting property against all cofibration and that lifting property against the $j_i$ is enough to rectify this in an actual lifting property. One then check by hand the cofibration, weak equivalence and the map with the lifting property against the $j_i$ forms a model structure, with the same cofirbation and weak equivalences as the one you started from, hence is the the same model structure.

I found it explicitly written out as corollary 3.2 of:

Valery Isaev, On fibrant objects in model categories, Theory and Applications of Categories, Vol. 33, 2018, No. 3, pp 43-66, journal page, arXiv:1312.4327

Actually, you can, and you don't need accessibility (local presentability of the underlying category is enough).

Under your assumption, for each $i:A \to B$ a generating cofibration, take $B \coprod_A B \hookrightarrow I_A B \to B$ a cylinder object, and let $j_i : B \hookrightarrow I_A B$ be the first leg inclusion. Then the $j_i$ form a generating set of trivial cofibrations.

The proof essentially use that all objects are fibrant, and that you can consider the weak factorization system generated by the $j_i$ (so some smallness or local presentability assumption).

The key step in the proof is to observe that a map that has the RLP against the $j_i$ and is a weak equivalence is a trivial fibration. This easily follows from the fact that weak equivalences between fibrant objects have the "up to homotopy" lifting property against all cofibration and that lifting property against the $j_i$ is enough to rectify this in an actual lifting property. One then check by hand the cofibration, weak equivalence and the map with the lifting property against the $j_i$ forms a model structure, with the same cofirbation and weak equivalences as the one you started from, hence is the the same model structure.

I found it explicitly written out as corollary 3.2 of this paper of Valery Isaev.

Actually, you can, and you don't need accessibility (local presentability of the underlying category is enough).

Under your assumption, for each $i:A \to B$ a generating cofibration, take $B \coprod_A B \hookrightarrow I_A B \to B$ a cylinder object, and let $j_i : B \hookrightarrow I_A B$ be the first leg inclusion. Then the $j_i$ form a generating set of trivial cofibrations.

The proof essentially use that all objects are fibrant, and that you can consider the weak factorization system generated by the $j_i$ (so some smallness or local presentability assumption).

The key step in the proof is to observe that a map that has the RLP against the $j_i$ and is a weak equivalence is a trivial fibration. This easily follows from the fact that weak equivalences between fibrant objects have the "up to homotopy" lifting property against all cofibration and that lifting property against the $j_i$ is enough to rectify this in an actual lifting property. One then check by hand the cofibration, weak equivalence and the map with the lifting property against the $j_i$ forms a model structure, with the same cofirbation and weak equivalences as the one you started from, hence is the the same model structure.

I found it explicitly written out as corollary 3.2 of this paper of Valery Isaev.

Actually, you can, and you don't need accessibility (local presentability of the underlying category is enough).

Under your assumption, for each $i:A \to B$ a generating cofibration, take $B \coprod_A B \hookrightarrow I_A B \to B$ a cylinder object, and let $j_i : B \hookrightarrow I_A B$ be the first leg inclusion. Then the $j_i$ form a generating set of trivial cofibrations.

The proof essentially use that all objects are fibrant, and that you can consider the weak factorization system generated by the $j_i$ (so some smallness or local presentability assumption).

I found it explicitly written out as corollary 3.2 of this paper of Valery Isaev.

Actually, you can, and you don't need accessibility (local presentability of the underlying category is enough).

Under your assumption, for each $i:A \to B$ a generating cofibration, take $B \coprod_A B \hookrightarrow I_A B \to B$ a cylinder object, and let $j_i : B \hookrightarrow I_A B$ be the first leg inclusion. Then the $j_i$ form a generating set of trivial cofibrations.

The proof essentially use that all objects are fibrant, and that you can consider the weak factorization system generated by the $j_i$ (so some smallness or local presentability assumption).

The key step in the proof is to observe that a map that has the RLP against the $j_i$ and is a weak equivalence is a trivial fibration. This easily follows from the fact that weak equivalences between fibrant objects have the "up to homotopy" lifting property against all cofibration and that lifting property against the $j_i$ is enough to rectify this in an actual lifting property. One then check by hand the cofibration, weak equivalence and the map with the lifting property against the $j_i$ forms a model structure, with the same cofirbation and weak equivalences as the one you started from, hence is the the same model structure.

I found it explicitly written out as corollary 3.2 of this paper of Valery Isaev.