For a Hopf algebra $H$, denote by ${}_H\!{\rm YD}$ the Yetter-Drinfelʹd category associated to $H$. If a coalgebra $C$ is a Yetter-Drinfelʹd module over $H$ and its structural maps are morphisms in ${}_H\!{\rm YD}$, one calls $C$ a coalgebra in ${}_H\!{\rm YD}$. For a braided cocommutative coalgebra $C$ in ${}_H\!{\rm YD}$, the authors prove that the left $C$-comodule category ${}^C_H\!{\rm YD}$ is a tensor category and that a braiding structure in ${}_H\!{\rm YD}$ induces a braiding structure in ${}^C_H\!{\rm YD}$ if and only if $\Psi_{N,C}\circ \Psi_{C,N}\circ {}^C\!\Gamma_N={}^C\!\Gamma_N$ for any object $N$ in ${}^C_H\!{\rm YD}$. In particular, assume that $H$ is a braided Hopf algebra with an invertible antipode and $C$ is a braided cocommutative coalgebra in the $H$-comodule category ${}^H\!M$; then a braiding structure in ${}^H\!M$ induces a braiding structure in ${}^{C\times H}\!M$ if and only if $\Psi_{N,C}\circ \Psi_{C,N}\circ {}^C\!\Gamma_N={}^C\!\Gamma_N$ for any object $N$ in ${}^{C\times H}\!M$, where $C\times H$ denotes the smash coproduct of $C$ and $H$.