Let $A$ be an Artin algebra and $C$ a subcategory of mod-$A$ that contains all projective modules and is closed under finite direct sums (but not necessarily under direct summands).
Let $T:=add(C)$.


For an $A$-module $M$, let $f: X \rightarrow M$ be a right $C$-approximation with $X$ in $C$. Then it sounds plausible that a minimal right $T$-approximation $g: Y \rightarrow M$ has the property that $Y$ is a direct summand of $X$, since this is characterised by $Hom(Z,g)$ being epi for all $Z \in add(C)$ (and C and add(C) are the same up to multiplicity of summands).


But since I get a strange consequence for this in a special case (and I never really worked with subcategory that are not closed under direct summands), I wanted to ask about this here.



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> Question: Is there a reference for this? (is it even correct?)
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Right approximation in certain subcategories
I believe this is true. By Proposition 5.1.2 of *Relative Homological Algebra* by Enochs and Jenda, a minimal right $T$-approximation will be a direct summand of any right $T$-approximation. 


Therefore it suffices to show that $f:X\to M$ is also a right $T$-approximation. Let $h:\tilde{Y}\to M$ be any morphism with $\tilde{Y}\in T$. Then there is a $Z\in T$ such that $\tilde{Y}\oplus Z\in C$ and nonzero map $h\circ\pi :\tilde{Y}\oplus Z\to M$. Since $f:X\to M$ is a right $C$-approximation, there is an $\alpha:\tilde{Y}\oplus Z\to X$ such that $f\circ\alpha = h\circ\pi$. This clearly restricts to a non-zero map $\alpha\vert:\tilde{Y}\to X$ such that $g = f\circ\alpha\vert$. In other words, $f:X\to M$ is a right $T$-approximation.


