In a model category $\mathscr M$ (in the modern sense, i.e. closed and with functorial factorizations), there is a notion of fibrant and cofibrant replacement functors.
Specifically, for any object $x$ of $\mathscr M$, the $(\mathrm{Cofib}\cap W, \mathrm{Fib})$-factorization $$ x \to x' \to 1 \qquad \text{($1$ final object)}$$ gives rise to a fibrant replacement functor $R \colon \mathscr M \to \mathscr M$. Dually, the $(\mathrm{Cofib}, \mathrm{Fib}\cap W)$-factorization $$ 0 \to x'' \to x \qquad \text{($0$ initial object)} $$ gives rise to a cofibrant replacement functor $Q \colon \mathscr M \to \mathscr M$.
I did not choose the letter $Q$ and $R$ randomly. They are all over the literature (Hovey, Goerss-Jardine, etc.). Why those letters? They hardly seem natural...