Let $U:\mathscr{A}\rightarrow\mathscr{B}$ and $F:\mathscr{B}\rightarrow\mathscr{A}$ be two functors. Suppose there is a natural transformation $\eta:\mathbf{1}_\mathscr{B} \rightarrow UF$ such that satisfies the following property: for every object $X$ in $\mathscr{B}$, for every object $Y$ in $\mathscr{A}$, and for every morphism $f:X\rightarrow UY$ in $\mathscr{B}$ there is a unique morphism $g:FX\rightarrow Y$ in $\mathscr{A}$ such that
$$ Ug \circ \eta_X = f $$
(where $\eta_X$ is the component of $\eta$ at $X$)
Now suppose that $\eta$ is also a natural isomorphism, can I claim that $(F, U, \eta, \epsilon)$, where $\epsilon$ is a counit of the adjunction, is an adjoint equivalence?