Yes this is true. Specifically:
Proposition. Let $L:\mathcal B\to\mathcal A$ be left adjoint to $R:\mathcal A\to\mathcal B$ where $\mathcal A$ and $\mathcal B$ are additive categories. Then $L$ and $R$ are additive.
Proof. Recall that $F:\mathcal A\to\mathcal B$ is additive if and only if $F$ preserves binary products (prove this!). But $R$ is right adjoint so $R$ preserves limits. In particular, $R$ preserves binary products.
Proving $L$ is additive is similar once combined with the observation that binary products coincide with binary coproducts in an additive category. $\Box$
Of course, if you don't like using continuity properties of adjoint functors, you could directly prove the proposition. This is a nice exercise too.
Side-note. This is a useful result in geometry as a morphism $f:X\to Y$ of spaces defines functors $\DeclareMathOperator{Sh}{Sh}$
\begin{align*}
f_* &:\Sh(X)\to\Sh(Y) \\
f^{-1} &: \Sh(Y)\to\Sh(X)
\end{align*}
One shows that $f^{-1}$ is left adjoint to $f_*$ and the above ensures the additivity of both.
In fact, one may further prove that right adjoint functors between abelian categories are left exact and left adjoint functors between abelian categories are right exact.
Let's say that a monoidal subcategory $Z$ of a monoidal category $(X,\otimes,I)$ is one for which
- Morphisms $f,g\in Z$ imply $f\otimes g\in Z$
- Objects $A,B,C\in Z$ imply the associators $(A\otimes B)\otimes C\to A\otimes(B\otimes C)$ are in $Z$
- $I\in Z$
- An object $A\in Z$ implies the unitors $I\otimes A\cong A\cong A\otimes I$ are in $Z$.
Evidently the intersection of families of monoidal subcategories is a monoidal subcategory. Therefore, given a functor $F\colon Y\to X$ there is a smallest monoidal subcategory $MF(Y)$ containing the image of $F(Y)$. In particular, $F\colon Y\to X$ factors as $Y\to MF(Y)\hookrightarrow X$. Moreover, $MF(Y)\hookrightarrow X$ is a monoidal functor, i.e. a functor in the category of small monoidal categories.
I now claim the morphisms in this subcategory are exactly the composites of morphisms of the form $F(f_1)\otimes F(f_2)\otimes...\otimes F(f_n)$ (wih various parenthesizations) for $f_i$ morphisms in $X$, and appropriate unitors and associators.
It follows that $MF(Y)$ has cardinality bounded by $\kappa_Y$, where $\kappa_Y$ is the smallest infinite cardinal bounding the cardinality of $Y$.
If $\lambda$ is its cardinal, then the isomorphism $\lambda\cong UM(Y)$ of $\lambda$ with the set of morphisms of $MF(Y)$ induces a monoidal category structure on $\lambda$ so that the resulting monoidal category is isomorphic to $MF(Y)$.
Thus the functor $F\colon Y\to X$ factors as $Y\to M\to X$ where $M\to X$ is a monoidal functor, and $UM$ is a cardinal bounded by $\kappa_Y$. Since the set of cardinals bounded by $\kappa_Y$ is a set, and since each set has a set of monoidal structures, and since between any two categories there is a set of functors between them, it follows that for every category $Y$ there is only a set of functors $Y\to M$ where $M$ is a monoidal category with $UM$ a cardinal bounded by $\kappa_Y$.
By the previous discussion, this is a solution set for the forgetful functor from small monoidal categories to small categories: any functor $F\colon Y\to X$ factors as $Y\to M\to X$ where $M\to X$ is a monoidal functor with $UM$ a cardinal bounded by $\kappa_Y$.
Best Answer
I think my comments have turned into an answer. The property of the question is equivalent to being a "SAFT category" in the terminology of the question, a.k.a. to being "compact" in the sense of Isbell. A reference for this notion is Borger, Tholen, Wischnewsky, and Wolff, which also gives several theorems that imply that $Top$ is Isbell-compact and compares to some of the other notions listed in the question. They also show, for example, that some of these notions coincide under (co)generation hypotheses.
I believe there may be other papers of Borger which compare this notion to totality.