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This is discussed in section 3.4.3 of https://arxiv.org/abs/2004.00731 by Mathieu Anel and Chaitanya Leena Subramaniam.

Here's an argument which I currently believe. As Tim suggested, it does use the fat small object argument. References are to that paper. Let $\mathcal{K}$ be a locally $\lambda$-presentable category …

Remark 1.30 of Adámek and Rosický, Locally Presentable and Accessible Categories claims that in any locally $\lambda$-presentable category, each $\mu$-presentable object (for $\mu\ge\lambda$) can be w …

Let $M$ be a model category. I don't assume that $M$ has functorial factorizations or that $M$ is simplicial. Write $M^{c}$ (respectively, $M^{cf}$) for the full subcategory of $M$ on the cofibrant ob …

When $\mathcal{M}$ and $\mathcal{N}$ are combinatorial, this class $\mathcal{C}$ is indeed the left class of a weak factorization system on the category of all left adjoints from $\mathcal{M}$ to $\ma …

I would go even farther than the comments above, at least in the specific case you mention about computing (co)limits objectwise in a functor category. Once you know the statement for limits, deducin …

I'm not sure exactly what you're looking for, but here is a somewhat weird example of a pushout in commutative monoids I recently came across: $\begin{matrix} \mathbb{N}&\to&\mathbb{Z}\\\\ \downarrow …