A $\mathbb Z_p$-algebra homomorphism from $\mathbb Z_p[[T]]$ determined by its value on $1+T$ (?)

Question

Let $f$ and $g$ be $\mathbb Z_p$-algebra homomorphisms from $\mathbb Z_P[[T]]$ to $\varprojlim\limits_{n} \mathbb Z_p[\Gamma/\Gamma^{p^n}]$, where $\Gamma$ is the abelian group $\mathbb Z_p$ written multiplicatively ($\Gamma^{p^n}$ corresponds to $p^n\mathbb Z_p$). Is it true that if $f$ and $g$ agree on $1+T$, they are equal? I need the result when $f$ and $g$ are bijective.