Let $\mathcal{C}$ be a small category and let $\text{sPre}(\mathcal{C})$ be the category of simplicial presheaves on $\mathcal{C}$, i.e. $\text{sPre}(\mathcal{C})=\mathbf{sSet}^{\mathcal{C}^{\text{op}}}$. Let us denote by $U\mathcal{C}$ the model category obtained by endowing $\text{sPre}(\mathcal{C})$ with the projective model structure (so that weak equivalences or fibrations in $U\mathcal{C}$ are the natural transformations of simplicial presheaves which are sectionwise weak equivalences or fibrations for the Kan-Quillen model structure on $\mathbf{sSet}$).
Question 1: Can we provide an example of a small category $\mathcal{C}$ for which $U\mathcal{C}$ has the following property: there is a fibration $f\colon X\rightarrow Y$ and a trivial cofibration $i\colon A\rightarrow Y$ in $U\mathcal{C}$ such that the pulled back map $f^*(i)\colon X\times_{Y}A\rightarrow X$ is not a projective cofibration?
Question 2: Are there (interesting) sufficient conditions on $\mathcal{C}$ which automatically ensure that $U\mathcal{C}$ has the property of Question 1?
In general, these questions seem a bit tricky to approach since we do not have a very good descriptions of (trivial) cofibrations for the projective model structure, but maybe there are some easy/obvious/trivial cases I am missing.
Any comment/idea would be very much appreciated. Thanks in advance.