  This paper is devoted to the study of Hopf braces projections in a monoidal
setting. Given a cocommutative Hopf brace ${\mathbb H}$ in a strict symmetric
monoidal category ${\sf C}$, we define the braided monoidal category of left
Yetter-Drinfeld modules over ${\mathbb H}$. For a Hopf brace ${\mathbb A}$ in
this category, we introduce the concept of bosonizable Hopf brace and we prove
that its bosonization ${\mathbb
A}\blacktriangleright\hspace{-0.15cm}\blacktriangleleft {\mathbb H}$ is a new
Hopf brace in ${\sf C}$ that gives rise to a projection of Hopf braces
satisfying certain properties. Finally, taking these properties into account,
we introduce the notions of v$_{i}$-strong projection over ${\mathbb H}$,
$i=1,2,3,4$, and we prove that there is a categorical equivalence between the
categories of bosonizable Hopf braces in the category of left Yetter-Drinfeld
modules over ${\mathbb H}$ and the category of v$_{4}$-strong projections over
${\mathbb H}$.
