$ \newcommand{\cat}[1]{\mathcal{#1}} $ Let $ (\cat C,J) $ be a site. I'm trying to show that if the topology $ J $ comes from a pretopology $ K $ in the sense that for each $ X\in \cat C $ and for each covering sieve $ S\in J(X) $, there exists some covering $ k = \{U_i\xrightarrow{\phi_i} X\}_{i\in I}\in K(X) $ of $ X $ such that $ k\in S $, then the $ J $-sheaves are exactly the $ K $-sheaves.
I recall some definitions. A presheaf $ F\colon \cat C^\mathrm{op}\to \mathrm{Set} $ is a $ J $-sheaf if for any $ X\in \cat C $, for any covering sieve $ S\in J(X) $ and for any matching family $ \sigma\colon S\to F $, there exists a unique morphism of presheaves $ h_X\to F $ such that the obvious diagram commutes, where $ h_X = \cat C({-},X) $. On the other hand, $ F $ is a $ K $-sheaf if for each $ X\in \cat C $, for each covering $ \{U_i\xrightarrow{\phi_i} X\}_{i\in I}\in K(X) $ and for each family of sections $ s_{\phi_i}\in F(U_i) $ such that the obvious compatibility condition holds, there exists a unique $ s\in F(X) $ that is compatible with the $ s_\phi $s once "restricted".
To show the implication "$ J $-sheaf implies $ K $-sheaf" I noticed that I need the following condition to hold: given an object $ X\in \cat C $ and a covering $ k = \{U_i\xrightarrow{\phi_i} X\}_{i\in I}\in K(X) $, the sieve $ S_k $ of all the arrows $ U\xrightarrow f X $ that factors through some $ \phi $ is again a $ J $-covering sieve.
Is it always the case? Or is it the case that the theorem I'm trying to prove is false?