  Let $X$ be a ball quasi-Banach function space on ${\mathbb R}^n$ satisfying
some mild assumptions and let $\alpha\in(0,n)$ and $\beta\in(1,\infty)$. In
this article, when $\alpha\in(0,1)$, the authors first find a reasonable
version $\widetilde{I}_{\alpha}$ of the fractional integral $I_{\alpha}$ on the
ball Campanato-type function space $\mathcal{L}_{X,q,s,d}(\mathbb{R}^n)$ with
$q\in[1,\infty)$, $s\in\mathbb{Z}_+^n$, and $d\in(0,\infty)$. Then the authors
prove that $\widetilde{I}_{\alpha}$ is bounded from
$\mathcal{L}_{X^{\beta},q,s,d}(\mathbb{R}^n)$ to
$\mathcal{L}_{X,q,s,d}(\mathbb{R}^n)$ if and only if there exists a positive
constant $C$ such that, for any ball $B\subset \mathbb{R}^n$,
$|B|^{\frac{\alpha}{n}}\leq C \|\mathbf{1}_B\|_X^{\frac{\beta-1}{\beta}}$,
where $X^{\beta}$ denotes the $\beta$-convexification of $X$. Furthermore, the
authors extend the range $\alpha\in(0,1)$ in $\widetilde{I}_{\alpha}$ to the
range $\alpha\in(0,n)$ and also obtain the corresponding boundedness in this
case. Moreover, $\widetilde{I}_{\alpha}$ is proved to be the adjoint operator
of $I_\alpha$. All these results have a wide range of applications.
Particularly, even when they are applied, respectively, to Morrey spaces,
mixed-norm Lebesgue spaces, local generalized Herz spaces, and mixed-norm Herz
spaces, all the obtained results are new. The proofs of these results strongly
depend on the dual theorem on $\mathcal{L}_{X,q,s,d}(\mathbb{R}^n)$ and also on
the special atomic decomposition of molecules of $H_X(\mathbb{R}^n)$ (the
Hardy-type space associated with $X$) which proves the predual space of
$\mathcal{L}_{X,q,s,d}(\mathbb{R}^n)$.
