Confirmation needed of the fact that subcategory $\mathbf{Lat}$ is not full in $\mathbf{Pos}$

If you are familiar with this stuff then you probably don't need the information I have added. So let me start with the question:

Can you prove that category $$\mathbf{Lat}$$ is not a full subcategory of category $$\mathbf{Pos}$$?

I dived into the theory of universal algebras and encountered lattices. I noticed that a lattice $$\left(L,\wedge,\vee\right)$$ can be desribed as a poset that is equipped with binary products and coproducts. Here $$a\leq b$$ is defined by $$a=a\wedge b$$. Then $$a\sqcap b=a\wedge b$$ and $$a\sqcup b=a\vee b$$. There is a category $$\mathbf{Lat}$$ having lattices as objects and functions that respect $$\wedge$$ and $$\vee$$ as arrows. These functions are order-preserving which makes them arrows in $$\mathbf{Pos}$$ having posets as objects and orderpreserving functions as arrows. So $$\mathbf{Lat}$$ is a subcategory of $$\mathbf{Pos}$$ but I 'know' that it is not a full subcategory. As a confirmation of that I am looking for an arrow in $$\hom_{\mathbf{Pos}}\left(L,L'\right)$$ that is not an arrow in $$\hom_{\mathbf{Lat}}\left(L,L'\right)$$. Here $$L$$ and $$L'$$ are lattices. So it is about a function that is orderpreserving, but does not respect (at least one of) the operations $$\wedge,\vee$$.

$$\begin{matrix} & & D & & & & & & D \\ & \nearrow & & \nwarrow & & & & \nearrow & & \nwarrow \\ B & & & & C & \quad \Longrightarrow \quad & B & & & & C & \\ & \nwarrow & & \nearrow & & & & \nwarrow & & \nearrow \\ & & A & & & & & & E \\ & & & & & & & & \uparrow \\ & & & & & & & & A \end{matrix}$$
• $A=B\sqcap C$ and $fA\neq fB\sqcap fC=E$ right? Thank you. Commented Jan 13, 2014 at 13:38