Let $C$ be a symmetric monoidal category, and $f : x \to y$ be a morphism in $C$. I would like to construct the localization $C_f$ explicitly, which solves the universal property $$\mathrm{Hom}_{\otimes}(C_f,D) = \{F \in \mathrm{Hom}_{\otimes}(C,D) : F(f) \text{ iso}\}.$$

I am not interested in a general existence proof or alike; instead I would like to exhibit $C_f$ as an *explicit* full reflective subcategory of $C$, thereby also showing its existence. The reason is that I want to actually compute something in these localizations which a priori does not simply follow from the universal property.

There is a general construction of a localization of a *plain category* with respect to arbitrary sets of morphisms, which can be found on page 6 of Gabriel-Zisman's Calculus of fractions and homotopy theory. Thus, an object of the localization $C_f$ is an object of $C$, and a morphism is a class of a finite sequence of the form $f_1 s^{-1} f_2 \cdots s^{-1} f_n$ (perhaps without $f_1$ or $f_n$), where the sources and targets should fit, subject to the obvious cancellation rules. But this might not define a (small) set of morphisms, right?

**Question 1.** Which conditions have to be imposed on $C$ and $f$ so that $C_f$ exists (without leaving the universe)?

I know the basics about left/right multiplicative systems (as in Gabriel-Zisman, Weibel, Kashiwara-Schapira, etc.), saturations etc., but I could not find any answer to this question in the literature.

EDIT: As Theo points out, there is no set-theoretic problem if we localize a category at just one single morphism. But when $C$ is monoidal, there is no reason why the tensor product $C \times C \to C$ extends to a tensor product $C_f \times C_f \to C_f$, because this means that for every $x \in C$ the invertibility of $f$ forces the invertibility of $x \otimes f$ and $f \otimes x$ in the language of categories, which is unplausible. Instead we should better localize at all morphisms $x \otimes f$ and $f \otimes x$, where $x$ runs through all objects of $C$; this is a *monoidal class* in the language of Day's paper "A Note on Monoidal Localisation". But now there *are* set-theoretic problems in the description of $C_f$ above. On the other hand, we repair this easily if $C$ has a small colimit-dense subcategory, which happens to be the case when $C$ is presentable.

**Question 2.** Which conditions have to be imposed on $C$ and $f$ so that we can write down explicitly the localization $C_f$ in the $2$-category of symmetric monoidal categories? How does it look like?

**Question 3.** Actually I am interested in *cocomplete* symmetric monoidal categories and *cocontinuous* symmetric monoidal functors between them. How does the localization look like in this context?

Let me mention a special case where everything works out: Let $\mathcal{L} \in C$ be an object whose symmetry $\mathcal{L}^{\otimes 2} \to \mathcal{L}^{\otimes 2}$ is the identity and $f : 1_C \to \mathcal{L}$ a morphism (imagine a global section of a line bundle on a scheme). Let $C_f \subseteq C$ the full subcategory consisting of those $M \in C$ such that $M \otimes f : M \to M \otimes \mathcal{L}$ is an isomorphism. If $C$ is cocomplete, the inclusion $C_f \subseteq C$ has a left adjoint: It maps $M \in C$ to the colimit of $M \to M \otimes \mathcal{L} \to M \otimes \mathcal{L}^{\otimes 2} \to \dotsc$. Using this left adjoint, one can define tensor products and colimits in $C_f$ (one may cite Day's reflection theorem here) and verify easily that $C \leadsto C_f$ is the localization in the context of cocomplete symmetric monoidal categories.

One can also verify that if $\mathcal{L}$ is a line bundle on a scheme $X$ and $f \in \Gamma(X,\mathcal{L})$ is a global section, then we really have $\mathrm{Qcoh}(X)_f = \mathrm{Qcoh}(X_f)$, so this categorical localization is compatible with the scheme theoretic localization.

However, for other applications, I need more general morphisms $f$. This motivated my question. I am pretty sure that this should be standard in category theory, therefore the reference request tag.

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