It is known (follows for example from Proposition 4.2 of Simplicial Homotopy Theory by Goerss and Jardine) that a Kan-fibration can be defined as a map having the right lifting property with respect to all maps of the form $$(\Delta^1\times\partial\Delta^n)\coprod_{\Delta^{\{i\}}\times\partial\Delta^n}(\Delta^{\{i\}}\times\Delta^n)\hookrightarrow \Delta^1\times\Delta^n$$ for $n\geq 0,\:i=0,1$.

Moving to topological spaces, a Serre-fibration is defined to be a map having the right lifting property with respect to all maps of the form $$([0,1]\times\partial D^n)\coprod_{\{0\}\times\partial D^n}(\{0\}\times D^n)\hookrightarrow [0,1]\times D^n$$ for $n\geq 0$. This resembles the definition of Kan-fibration given above, but we need not take $i=0,1$ as in the simplicial case, because of the inherent symmetry of topological spaces. Now in topological spaces, the above maps are isomorphic (homeomorphic) to the maps
$$\{0\}\times D^n\hookrightarrow [0,1]\times D^n,$$ so a Serre-fibration can be defined as a map having the right lifting property with respect to this last class of maps. Taking into account the nonsymmetry of simplicial sets, my question is

Can we define a Kan fibration to be a map having the right lifting property with respect to all maps of the form $$\Delta^{\{i\}}\times\Delta^n\hookrightarrow \Delta^1\times\Delta^n$$ for $n\geq 0,\:i=0,1$?

It is known (follows for example from Proposition 4.2 of Simplicial Homotopy Theory by Goerss and Jardine) that a Kan-fibration can be defined as a map having the right lifting property with respect to all maps of the form $$(\Delta^1\times\partial\Delta^n)\coprod_{\Delta^{\{i\}}\times\partial\Delta^n}(\Delta^{\{i\}}\times\Delta^n)\hookrightarrow \Delta^1\times\Delta^n$$ for $n\geq 0,\:i=0,1$.

Moving to topological spaces, a Serre-fibration is defined to be a map having the right lifting property with respect to all maps of the form $$([0,1]\times\partial D^n)\coprod_{\{0\}\times\partial D^n}(\{0\}\times D^n)\hookrightarrow [0,1]\times D^n$$ for $n\geq 0$. This resembles the definition of Kan-fibration given above, but we need not take $i=0,1$ as in the simplicial case, because of the inherent symmetry of topological spaces. Now in topological spaces, the above maps are isomorphic (homeomorphic) to the maps
$$\{0\}\times D^n\hookrightarrow [0,1]\times D^n,$$ so a Serre-fibration can be defined as a map having the right lifting property with respect to this last class of maps. Taking into account the nonsymmetry of simplicial sets, my question is

Can we define a Kan fibration to be a map having the right lifting property with respect to all maps of the form $$\Delta^{\{i\}}\times\Delta^n\hookrightarrow \Delta^1\times\Delta^n$$ for $n\geq 0,\:i=0,1$?