A simplicial model category satisfies the axiom
SM7: if for any cofibration $i:A\to X$ and any fibration $p:E\to B$ the map of simplicial sets (induced by the functoriality of map) $map_M(X,E)\to map_M(X,E)\times_{map_M(A,B)}map_M(X,B)$ is a fibration of simplicial sets which is moreover a weak equivalence if either $i$ or $p$ is.
Here is exercise 3.17 from A primer for unstable motivic homotopy theory :
Let $M$ be a simplicial model category, $A\to X$ is a cofibration, then for any object $Y$, show that the natural map $map_M(Y,A)\to map_M(Y,X)$ is a fibration of simplicial set.
I know that for $A\to X$ a fibration, it holds for the category of simplicial sets and the proof for simplicial model category should be similiar. But I'm not sure how to prove the exercise where $A\to X$ is a cofibration.
This is false. For instance, if $M$ is just simplicial sets with their usual simplicial model category structure and you take $Y$ to be a point, then $\operatorname{map}_M(Y,X)$ is naturally isomorphic to $X$ so this would be saying that every cofibration $A\to X$ is also a fibration.