(**Edit:** the argument below actually leads to a *positive* answer, if the arrows in the category of lctvs are considered to be 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 if one define the arrows of the former to be continuous linear maps, which is standard for (linear) functional analysis.

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