Are there dual logic gates? In category theory the usefulness of dualising such devices as monoids to comonoids has been shown, where multiplication which takes two inputs to one output is dualised to comultiplication which takes one input and has two outputs.
Is there a similar interpretation for logic? Can there be such a thing as a co-and or a co-or gate?
 A: This is an interesting question. Below are my thoughts. 
Logic gates as monoids
The $\vee$ (or) and $\wedge$ (and) logic gates are monoids over the set of boolean values $\mathbb{B} = \{F, T\}$ (false and true): $(\mathbb{B}, \vee, T)$ and $(\mathbb{B}, \wedge, F)$ respectively. One way to think of this categorically is as a monoid object in some monoidal category. Since we are dealing with sets, let this category be the category of sets with the monoidal structure of Cartesian products with some arbtirary singleton set 1 = $\{\bullet\}$ i.e.  $(\mathbf{Set}, \times, 1)$. Thus, the multiplication and unit operations of the or and and monoids are $\vee : \mathbb{B} \times \mathbb{B} \rightarrow \mathbb{B}$, $T : 1 \rightarrow \mathbb{B}$ and $\wedge : \mathbb{B} \times \mathbb{B} \rightarrow \mathbb{B}$, $F : 1 \rightarrow \mathbb{B}$ respectively. 
Dualising to a comonoid
Comonoid objects over $\mathbb{B}$ in $\mathbf{Set}$ have a comultiplication $\delta : \mathbb{B}  \rightarrow \mathbb{B} \times \mathbb{B}$ and counit $\varepsilon : \mathbb{B} \rightarrow 1$. So, as you say, one input is mapped to two outputs. However, the comonoid laws (dual to the monoid laws) state that $(id \times \varepsilon) \circ \delta = \langle{id, !_\mathbb{B}}\rangle$ and $(\varepsilon \times id) \circ \delta = \langle{!_\mathbb{B}, id}\rangle$ where $!_A : A \rightarrow 1$ i.e. comultiplication of an input followed by a counit of one component is the same as just pairing that input with some singleont set (reverse the arrows in the diagram for the monoid laws). In $\textbf{Set}$ this means the only option for $\delta$ (to satisfy the laws) is duplication, i.e., $\delta = \langle{id, id}\rangle$ ($\delta\ x \mapsto (x, x)$). Therefore  co-or and co-and  can only ever be duplication, with the counit $\varepsilon = !_\mathbb{B}$ mapping its input to the arbitrary singleton set 1 (forgetting its input). There is no other choice. 
Relation between $\vee$/$\wedge$ monoids and the duplication comonoid 
The duplication operation is the right-identity (section) of the $\vee$ and $\wedge$ operations i.e. $\vee \circ \delta = id$ and $\wedge \circ \delta = id$ (since $\vee$ and $\wedge$ are idempotent monoids). However, $\delta$ is not the left-inverse (retract), e.g., $\delta \circ \vee \neq id$, thus they do not form an isomorphism. This can be understood easily from an information-theoretic perspective since $\vee$ reduces the amount of information from two-bits to one-bit, and $\delta$ preserves the amount of information by duplicating the single bit, whereas $id$ preserves both bits of information. 
Note that the counit of this comonoid is the left-identity (retract) of $F$ and $T$ i.e. $\varepsilon \circ F = id$ and $\varepsilon \circ T = id$. The counit cannot be the right-identity (section). You can follow an information-theoretic argument similar to the above to understand why: $\varepsilon$ goes from 1-bit of information to 0-bits of information (since we always know what is in this singleton set $1$ anyway), thus no information can be carried on to the morphism $F : 1 \rightarrow \mathbb{B}$ and $T : 1 \rightarrow \mathbb{B}$, which create a  new bit of information e.g. $(T \circ \varepsilon) F \mapsto T$ thus $T \circ \varepsilon \neq id$.
Addendum 
Can we try to be more creative with our comonoid? For example, can we use a different monoidal structure over sets such as sets of pairs? Then, for example, we could define a dual to $\wedge$ that gives us all its "possible" inputs: $\delta\ F \mapsto [(F, F), (F, T), (T, F)]$ and $\delta\ T \mapsto [(T, T)]$. Thus the underlying monoidal category structure is $A \otimes B = \mathcal{P}(A \times B)$ where $f \otimes g\ \{(x_0, y_0), \ldots, (x_n, y_n)\} \mapsto \{(f\ x_0, g\ y_0), \ldots, (f\ x_n, g\ x_n)\}$. Unfortunately, this does not give us a comonoid as the laws are violated, e.g. $((\varepsilon \otimes id) \circ \delta)\ F \mapsto \{(\bullet, F), (\bullet, T)\}$ and $\langle{!_\mathbb{B}, id}\rangle F \mapsto \{(\bullet, F)\}$. I've tried to think of some other structures, but all meet a similar end.
