(**Edit:** the argument below actually leads to a *positive* answer, since the arrows in the category of lctvs are continuous linear maps, which are smooth in both the convenient and Michal-Bastiani sense!)

Recall that a map $\Phi:E\rightarrow F$ from a lctvs $E$ into another lctvs $F$ is said to be *Michal-Bastiani smooth* if its directional (Gâteaux) derivatives of order $k$

$$ D^k\Phi[x](y_1,\ldots,y_k) = \left.\frac{\partial^k}{\partial\lambda_1\cdots\partial\lambda_k}\right|_{\lambda_1=\cdots=\lambda_k=0}\Phi\left(x+\sum^k_{j=1}\lambda_j y_j\right)$$

exist and the maps $D^k\Phi:E\times E^k\rightarrow F$ are *(jointly) continuous* for all $k\in\mathbb{N}$. This notion of smoothness is the one used in Milnor's treatment of infinite-dimensional Lie groups (J. Milnor, "Remarks on Infinite-Dimensional Lie Groups". In: B. DeWitt, R. Stora, eds., Les Houches Session XL, *Relativity, Groups and Topology II* (North-Holland, 1984), pp. 1007-1057) and Hamilton's exposé of the Nash-Moser inverse function theorem (R.S. Hamilton, "The Inverse Function Theorem of Nash and Moser". [Bull. Amer. Math. Soc. **7** (1982) 65-222][1]).

Michal-Bastiani smooth maps are also smooth in the sense of Kriegl and Michor (i.e. composing with a smooth curve leads to another smooth curve; some call the latter *convenient smoothness*), since the chain rule holds for Michal-Bastiani smooth maps (the converse is not necessarily true, see below). In any case, since continuous linear maps are smooth in the Michal-Bastiani sense, the category of lctvs embeds into the category of diffeological spaces.

Beware, however: both notions of smoothness coincide for Fréchet spaces, but in general they differ: conveniently smooth maps need not even be continuous, whereas Michal-Bastiani smooth maps are always so. A counter-example to continuity for conveniently smooth maps may be found in the paper of H. Glöckner, "Discontinuous Non-Linear Mappings on Locally Convex Direct Products". Publ. Math. Debrecen **68** (2006) 1-13, [arXiv:math/0503387][2]. In particular, if neither $E$ nor $F$ are Fréchet spaces, there may be maps between $E$ and $F$ which are smooth in the diffeological sense and need not even be continuous, let alone Michal-Bastiani smooth.

  [1]: http://www.ams.org/journals/bull/1982-07-01/S0273-0979-1982-15004-2/S0273-0979-1982-15004-2.pdf
  [2]: http://arxiv.org/abs/math/0503387